Steady state of diffusion-advection on the torus

Let $P$ be a positive scalar function and $mathbf{v}(mathbf{x})$ is an assigned smooth vector field. The quantity $P(t,mathbf{x})$ evolves according to a transport equation of the kind
$$
partial_t P(mathbf{x},t) = -nabla cdot [ mathbf{v}(mathbf{x}) P(mathbf{x},t) – nabla P(mathbf{x},t) ]
$$
The steady state solution $P(mathbf{x})$ is given by
$$
nabla cdot [ mathbf{v}(mathbf{x}) P(mathbf{x}) – nabla P(mathbf{x}) ] = 0
$$
If $mathbf{v} = -nabla U $, then the formal solution has the usual Gibbs form
$$
P propto e^{-U }
$$
Question: assume that the above equation is defined on the 2D (flat) torus $mathbb{T}^2$.
How to deal with the case in which $mathbf{v} = -nabla U + mathbf{q}$, where $U$ has the periodicity imposed by $mathbb{T}^2$ and $mathbf{q} =(q_x,q_y)$ is a constant vector field?

More precisely, I’d like to find the class of solutions of
$$
nabla cdot [ mathbf{q} P(x,y) – P(x,y) nabla U(x,y) – nabla P(x,y) ] = 0
$$
with the periodic constraints typical of the flat torus $[0,1]times [0,1]$, i.e.

$P(0,y) = P(1,y)$, $quad P(x,0) = P(x,1)$,
$ qquad U(0,y) = U(1,y)$, $quad U(x,0) = U(x,1) , $.

If $mathbf{q} = 0$, then the solution $P propto e^{-U } $ works.
My feeling is that the problem with $mathbf{q} neq 0 $ is not trivial because of the topology of $mathbb{T}^2$: the constant field $mathbf{q}$ is clearly periodic (so it can live on the torus) but has no periodic potential (i.e. the potential should be $-q_x x -q_y y$ that is not periodic).

EDIT: I found this question about divergence-free fields on a torus. In fact, the field $mathbf{q} P(x,y) – P(x,y) nabla U(x,y) – nabla P(x,y)$ is required to be divergence-free.
Also these notes are interesting and deal with the diffusion on the flat torus (pag. 80).

In terms of differential forms the problem should be (correct me, I am not an expert):

$$J = q P -P dU – d P qquad qquad div(J)=0$$

where $J$ is a current 1-form, $q$ is a constant 1-form, $P$ and $U$ are 0-forms. In particular, we should have that $J = R dg$, where $g$ is a 0-form and $R$ is a 90-degrees rotation (i.e. it is the Hodge dual operator in two dimensions, $J = *dg$). Equivalently, by using a somewhat improper 3D terminology, $J = rot(A)$, where $A=(0,0,g)$. Maybe the language of differential forms helps understanding why it is not so easy to find a solution when the $mathbf{q}$ term is switched on.

Since this problem is related to systems studied in Physics (transport equations, Fokker-Planck equations), I posted a related question on physics SE.